The Large-Number Limit for Reaction Networks (Part 3)

We used to talk about reaction networks quite a lot here. When Arjun Jain was visiting the CQT, we made a lot of progress understanding how the master equation reduces to the rate equation in the limit where there are very large numbers of things of each kind. But we never told you the end of the story, and by now it’s been such a long time that you may need a reminder of some basic facts!

So…

The rate equation treats the number of things of each kind as continuous—a nonnegative real number—and says how it changes in a deterministic way.

The master equation treats the number of things of each kind as discrete—a nonnegative integer—and says how it changes in a probabilistic way.

You can think of the master equation as the ‘true’ description, and the rate equation as an approximation that’s good in some limit where there are large numbers of molecules — or more precisely, where the probability distribution of having some number of molecules of each kind is sharply peaked near some large value.

You may remember that in the master equation, the state of a chemical system is described by a vector in a kind of ‘Fock space’, while time evolution is described with the help of an operator on this space, called the ‘Hamiltonian’ :

The Hamiltonian is built from annihilation and creation operators, so all of this looks very much like quantum field theory. The details are here, and we won’t try to review them all:

The point is this: the ‘large-number limit’ where the master equation reduces to the rate equation smells a lot like the ‘classical limit’ of quantum field theory, where the description of light in terms of photons reduces to the good old Maxwell equations. So, maybe we can understand the large-number limit by borrowing techniques from quantum field theory!

How do we take the classical limit of quantum electromagnetism and get the classical Maxwell equations? For simplicity let’s ignore charged particles and consider the ‘free electromagnetic field’: just photons, described by the quantum version of Maxwell’s equations. When we take the classical limit we let Planck’s constant go to zero: that much is obvious. However, that’s not all! The energy of each photon is proportional to so to take the classical limit and get a solution of the classical Maxwell’s equations with nonzero energy we also need to increase the number of photons. We cleverly do this in such a way that the total energy remains constant as

So, in quantum electromagnetism the classical limit is also a large-number limit!

That’s a good sign. It suggests the same math will also apply to our reaction network problem.

But then we hit an apparent roadblock. What’s the analogue of Planck’s constant in chemical reaction networks? What should go to zero?

We told you the answer to that puzzle a while ago: it’s the reciprocal of Avogadro’s number!

You see, chemists measure numbers of molecules in ‘moles’. There’s a large number of molecules in each mole: Avogadro’s number. If we let the reciprocal of Avogadro’s number go to zero, we are taking a limit where chemistry becomes ‘continuous’ and the discreteness of molecules goes away. Of course this is just a mathematical trick, but it’s a very useful one.

So, we got around that roadblock. And then something nice happened.

When taking the classical limit of quantum electromagnetism, we focus attention on certain quantum states that are the ‘best approximations’ to classical states. These are called ‘coherent states’, and it’s very easy to study how the behave as we simultaneously let and let the expected number of photons go to infinity.

And the nice thing is that these coherent states are also important in chemistry! But because chemistry involves probabilities rather than amplitudes, they have a different name: ‘Poisson distributions’. On this blog, Brendan Fong used them to give a new proof of a great result in mathematical chemistry, the Anderson–Craciun–Kurtz theorem.

So, we have most of the concepts and tools in place, and we can tackle the large-number limit using quantum techniques.

As you’ll see, we solve the problem except for one important technical detail: passing a derivative through a limit! This means our main result is not a theorem. Rather, it’s an idea for how to prove a theorem. Or if we act like physicists, we can call it a theorem.

Review of notation

The rate equation says

where:

• is a vector describing concentrations of different species at time In chemistry these species could be different kinds of molecules.

• Each is a transition, or in chemistry, a reaction.

• is a vector of natural numbers saying how many items of each species appear as inputs to the reaction This is called the source of the reaction.

• is a vector of natural numbers saying how many items of each species appear as outputs of the reaction This is called the target of the reaction. So, says the net change of the number of items of each species in the reaction

• The rate at which the reaction occurs is proportional to the rate constant times the number

Here we are raising a vector to a vector power and getting a number, using this rule:

where is any vector of natural numbers and is any vector of nonnegative real numbers. From now on we’ll call a vector of natural numbers a multi-index.

• is the stochastic state saying the probability of having any particular number of items of each species at each time We won’t review the precise details of how this work; for that reread the relevant bit of Part 8.

• is the th number operator, defined using annihilation and creation operators as in quantum mechanics:

• Similarly, is the expected value of a certain product of operators. For any multi-index we define the falling power

and then we define

The large-number limit

Okay. Even if you don’t understand any of what we just said, you’ll see the master and rate equation look similar. The master equation implies this:

while the rate equation says this:

So, we will try to get from the first to the second second with the help of a ‘large-number limit’.

We start with a few definitions. We introduce an adjustable dimensionless quantity which we call This is just a positive number, which has nothing to do with quantum theory except that we’re using a mathematical analogy to quantum mechanics to motivate everything we’re doing.

Definition. The rescaled number operators are defined as This can be thought of as a rescaling of the number of objects, so that instead of counting objects individually, we count them in bunches of size

Definition. For any multi-index we define the rescaled falling power of the number operator by:

and also define

for any multi-index

Using these, we get the following equation:

where for any multi-index we set

This suggests a way to rescale the rate constants in the master equation:

Definition. The rescaled rate constants are

From here onwards, we change our viewpoint. We consider the rescaled rate constants to be fixed, instead of the original rate constants So, as we decrease we are studying situations where the original rate constants change to ensure that the rescaled rate constants stays fixed!

So, we switch to working with a rescaled master equation:

Definition. The rescaled master equation is:

This is really a one-parameter family of equations, depending on We write a solution of the rescaled master equation as but it is really one solution for each value of

Following the same procedure as above, we can rescale the rate equation, using the same definition of the rescaled rate constants:

Definition. The rescaled number of objects of the species is defined as where is the original number of objects of the species. Here again, we are counting in bunches of

Using this to rescale the rate equation, we get

Definition. The rescaled rate equation is

where

Therefore, to go from the rescaled master equation to the rescaled rate equation, we require that

as If this holds, we can identify with and get the rate equation from the master equation!

To this end, we introduce the following crucial idea:

Definition. A semiclassical family of states, is defined as a one-parameter family of states depending on such that for some we have

for every as

In particular, this implies

for every index

Intuitively, a semiclassical family is a family of probability distributions that becomes more and more sharply peaked with a larger and larger mean as decreases. We would like to show that in this limit, the rescaled master equation gives the rescaled rate equation.

We make this precise in the following propositions.

Proposition 1. If is a semiclassical family as defined above, then in the limit, we have as well.

Proof. For each index

By the definition of a semiclassical family,

since every term but the first approaches zero. Thus, we have

A similar but more elaborate calculation shows that

or in other words

as █

Proposition 2. If is a semiclassical family of states, then

for any multi-index

Proof. Consider the centered moment of the number operator:

Taking the limit as goes to zero, this becomes

For a general multi-index we can prove the same sort of thing with a more elaborate calculation. First note that

The right-hand side can be expanded as

We can write this more tersely as

where for any multi-index we define

and for any multi-indices we define

Now using the definition of a semiclassical state, we see

But this equals zero, as the last expression, expanded, is

where each individual sum is zero. █

Here is the theorem that would finish the job if we could give a fully rigorous proof:

“Theorem.” If is a solution of the rescaled master equation and also a semiclassical family for the time interval then is a solution of the rescaled rate equation for

Proof sketch. We sketch a proof that relies on the assumption that we can pass the limit through a time derivative. Of course, to make this rigorous, we would need to justify this. Perhaps it is true only in certain cases.

Assuming that we can pass the limit through the derivative:

and thus

and thus

As expected, we obtain the rescaled rate equation. █

Another question is this: if we start with a semiclassical family of states as our initial data, does it remain semiclassical as we evolve it in time? This will probably be true only in certain cases.

An example: rescaled coherent states

The best-behaved semiclassical states are the coherent states.
Consider the family of coherent states

using the notation developed in the earlier mentioned paper. In that paper it was shown that for any multi-index and any coherent state we have

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